Widely used quartz crystal based resonators can potentially be replaced by micromechanical, typically silicon-based, resonators in many applications. Silicon resonators can be made smaller than quartz resonators and there are a plurality standard manufacturing methods for silicon resonators. However, a problem associated with silicon based resonators is that they have a high temperature drift of the resonance frequency. The drift is mainly due to the temperature dependence of the Young modulus of silicon, which causes a temperature coefficient of frequency (TCF) approx. −30 ppm/C. This causes the resonance frequency to fluctuate due to changes in ambient temperature.
As concerns the TCF, both the linear, i.e. 1st order, and 2nd order behaviors are important in practice, since the first one represents local change of frequency on temperature change (ideally zero) and the second one, describing the curvature of the frequency vs. temperature curve, represents the width of the low-drift temperature range. If the first order term is zeroed, the frequency drift comes from the second order term alone, there being a certain “turnover temperature”, where the TCF achieves its absolute minimum value. The 2nd order TCF is herein also denoted TCF2 in contrast to 1st order coefficient TCF1 (linear TCF). AT-cut quartz crystals have near-zero low TCF1 and TCF2 at 25° C., their total frequency drift typically being within +−10 ppm over a wide temperature range of −40° C. . . . +85° C. (so-called industrial range). The temperature performance of silicon resonators is considerably worse at the present.
One promising approach to remove or mitigate the problem of temperature drift is extremely heavy doping of silicon. The effect of homogeneous n-type doping of concentration greater than 1019 cm−3 on bulk acoustic wave (BAW) resonator behavior has been discussed for example in WO 2012/110708. The document discusses that TCF1 of a “pure” c11-c12 mode (c11, c12 and c44 are elastic terms of the Young modulus of silicon) stays well above zero, and thus the frequency is still very dependent on temperature. However, other BAW resonance modes such as a square extensional (SE) or width extensional (WE) mode, have such dependence on elastic parameters c11, c12 (and c44), that the linear TCP can be made zero by correct selection of their in-plane geometry aspect ratio.
As concerns beam resonators, WO 2012/110708 teaches for example that 1st order TCP of beam resonators is minimized in extensional and flexural modes if the n-dopant concentration is between 1.6*1019 and 5*1019 cm−3 and the rotation angle in the wafer plane between 0 and 25 degrees from the [100] crystal direction in the (100) or (110) plane. Thus, there is a single point of temperature within these ranges at which there is no temperature drift of frequency. The document, however, does not teach how to achieve a broader stable temperature range of operation, i.e., how to minimize the 2nd order TCF. Also Jaakkola et al, “Determination of Doping and Temperature Dependent Elastic Constants of Degenerately Doped Silicon from MEMS Resonators”, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control Vol. 61, No. 7, pp 1063-1074, July 2014 suggests and shows by experiments that at least Lame plate resonators and length-extensional beam resonators can be effectively first order temperature compensated at an n-doping level greater than 2*1019 cm−3.
Another approach is to form an effective material structure with superimposed layers having different doping levels or crystal orientations, as discussed in U.S. Pat. No. 8,558,643. The structure forms a superlattice capable of carrying a resonance mode whose TCP is considerably less that of an undoped or homogeneously doped corresponding silicon element. Such structure can be also be used to decrease the 2nd order TCP to some extent so that temperature drift of less than 50 ppm over a 100° C. range is achieved.
The abovementioned documents cite also other documents utilizing silicon doping and briefly discusses also other methods to deal with the temperature drift problem.
The temperature behavior of a resonator is not only dependent on the doping concentration, but also on its geometry, crystal orientation and resonance mode excited therein, to mention some important factors. In addition, factors that need to be taken into account are the Q-value of the resonator, in which anchoring of the resonator plays an important role, and ability to manufacture the resonator design in practice. Low TCF and high Q-value may be contradictory design objectives using known resonator designs, since they are generally achieved with different geometrical layouts, for example.
At the present, there are only few practically feasible low-TCF silicon resonator designs available, some of which are disclosed in WO 2012/110708 and U.S. Pat. No. 8,558,643. However, there is a need for new and improved practically feasible designs, which allow for better control of TCF characteristics and simultaneously high Q-value. A simple structure and manufacturing process are also desirable.